Math Solver
Quadratic Solver
Solve any quadratic equation in standard form with real or complex roots, discriminant analysis, vertex details, and a live parabola graph. The full work stays visible so it still helps when you need to learn the process, not just the answer.
Equation Inputs
Enter coefficients for ax² + bx + c = 0. The leading coefficient `a` cannot be zero or the equation is not quadratic.
Ready. Enter coefficients and click Solve Equation.
x² - 5x + 6 = 0
1
Two real roots
x = 2.5
(2.5, -0.25)
(0, 6)
Parabola Graph
The graph auto-centers around the vertex and any real roots so you can see the curve shape immediately.
Parabola
Vertex
About this tool
Quadratic Solver
Quadratic Solver solves ax squared plus bx plus c equals zero, handling real and complex roots, showing the working step by step and graphing the parabola.
How to use it
- Enter the coefficients a, b and c.
- Read the roots and the step-by-step working.
- Check the graph to see where the parabola meets the x axis.
The discriminant tells you what to expect
The quadratic formula is x equals minus b, plus or minus the square root of b squared minus 4ac, all over 2a. The part under the square root, b squared minus 4ac, is the discriminant, and its sign determines everything before you finish the calculation.
Positive gives two distinct real roots and a parabola crossing the x axis twice. Zero gives one repeated root, with the parabola touching the axis at its vertex. Negative gives two complex conjugate roots, and the parabola never meets the axis at all.
Worked example
The default equation:
- Coefficientsa=1, b=-5, c=6
- Discriminant25 - 24 = 1
Result: Positive, so two real roots: x = 2 and x = 3.
When it helps
- Solving quadratics for homework or engineering work.
- Checking working step by step rather than just the answer.
- Seeing the relationship between the algebra and the graph.
- Finding where a projectile path crosses a height.
Common mistakes
- Forgetting the minus sign on b. The formula uses minus b, so a negative b becomes positive.
- Dividing only part of the numerator by 2a. The whole expression is divided.
- Assuming no real roots means no solution. It means the solutions are complex, which matters in engineering contexts.
